{"id":"https://openalex.org/W2098725746","doi":"https://doi.org/10.1017/s0963548306008376","title":"First-Order Definability of Trees and Sparse Random Graphs","display_name":"First-Order Definability of Trees and Sparse Random Graphs","publication_year":2007,"publication_date":"2007-01-23","ids":{"openalex":"https://openalex.org/W2098725746","doi":"https://doi.org/10.1017/s0963548306008376","mag":"2098725746"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548306008376","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548306008376","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A56C6BF6303133B88B09EB0630A9E845/S0963548306008376a.pdf/div-class-title-first-order-definability-of-trees-and-sparse-random-graphs-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A56C6BF6303133B88B09EB0630A9E845/S0963548306008376a.pdf/div-class-title-first-order-definability-of-trees-and-sparse-random-graphs-div.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079523589","display_name":"Tom Bohman","orcid":"https://orcid.org/0000-0001-5578-4664"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"TOM BOHMAN","raw_affiliation_strings":["Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail:","Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail:","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu)#TAB#","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056950019","display_name":"Alan Frieze","orcid":"https://orcid.org/0000-0002-8481-5615"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"ALAN FRIEZE","raw_affiliation_strings":["Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail:","Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail:","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu)#TAB#","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003433063","display_name":"Tomasz \u0141uczak","orcid":"https://orcid.org/0000-0002-3517-9034"},"institutions":[{"id":"https://openalex.org/I59411706","display_name":"Adam Mickiewicz University in Pozna\u0144","ror":"https://ror.org/04g6bbq64","country_code":"PL","type":"education","lineage":["https://openalex.org/I59411706"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"TOMASZ \u0141UCZAK","raw_affiliation_strings":["Department of Discrete Mathematics, Adam Mickiewicz University, Pozna\u0144 61-614, Poland (e-mail:","Department of Discrete Mathematics, Adam Mickiewicz University, Pozna\u0144 61-614, Poland (e-mail: tomasz@amu.edu.pl)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Discrete Mathematics, Adam Mickiewicz University, Pozna\u0144 61-614, Poland (e-mail:","institution_ids":["https://openalex.org/I59411706"]},{"raw_affiliation_string":"Department of Discrete Mathematics, Adam Mickiewicz University, Pozna\u0144 61-614, Poland (e-mail: tomasz@amu.edu.pl)#TAB#","institution_ids":["https://openalex.org/I59411706"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080095759","display_name":"Oleg Pikhurko","orcid":"https://orcid.org/0000-0002-9657-4011"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"OLEG PIKHURKO","raw_affiliation_strings":["Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail:","Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail:","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu)#TAB#","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076835963","display_name":"Clifford Smyth","orcid":"https://orcid.org/0000-0003-1486-7900"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"CLIFFORD SMYTH","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail:","Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail: csmyth@math.mit.edu)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail:","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail: csmyth@math.mit.edu)","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112765534","display_name":"Joel Spencer","orcid":null},"institutions":[{"id":"https://openalex.org/I36672615","display_name":"Courant Institute of Mathematical Sciences","ror":"https://ror.org/037tm7f56","country_code":"US","type":"education","lineage":["https://openalex.org/I36672615","https://openalex.org/I57206974"]},{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"JOEL SPENCER","raw_affiliation_strings":["Courant Institute, New York University, New York, NY 10012, USA (e-mail:","Courant Institute, New York University, New York, NY 10012, USA (e-mail: spencer@cims.nyu.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Courant Institute, New York University, New York, NY 10012, USA (e-mail:","institution_ids":["https://openalex.org/I36672615"]},{"raw_affiliation_string":"Courant Institute, New York University, New York, NY 10012, USA (e-mail: spencer@cims.nyu.edu)#TAB#","institution_ids":["https://openalex.org/I57206974"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064842516","display_name":"Oleg Verbitsky","orcid":"https://orcid.org/0000-0002-9524-1901"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"OLEG VERBITSKY","raw_affiliation_strings":["Institut f\u00fcr Informatik, Humboldt Universit\u00e4t Berlin, D-10099 Berlin, Germany (e-mail:","Institut f\u00fcr Informatik, Humboldt Universit\u00e4t Berlin, D-10099 Berlin, Germany (e-mail: verbitsk@informatik.hu-berlin.de)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Humboldt Universit\u00e4t Berlin, D-10099 Berlin, Germany (e-mail:","institution_ids":["https://openalex.org/I39343248"]},{"raw_affiliation_string":"Institut f\u00fcr Informatik, Humboldt Universit\u00e4t Berlin, D-10099 Berlin, Germany (e-mail: verbitsk@informatik.hu-berlin.de)#TAB#","institution_ids":["https://openalex.org/I39343248"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0639,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.75485526,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"16","issue":"3","first_page":"375","last_page":"400"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7207401990890503},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6920004487037659},{"id":"https://openalex.org/keywords/random-graph","display_name":"Random graph","score":0.677033543586731},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5356591939926147},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5259525775909424},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.49321621656417847},{"id":"https://openalex.org/keywords/quantifier","display_name":"Quantifier (linguistics)","score":0.49151572585105896},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4735385775566101},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4637297987937927},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1815795600414276},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06911405920982361}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7207401990890503},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6920004487037659},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.677033543586731},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5356591939926147},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5259525775909424},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.49321621656417847},{"id":"https://openalex.org/C2778523021","wikidata":"https://www.wikidata.org/wiki/Q5532504","display_name":"Quantifier (linguistics)","level":2,"score":0.49151572585105896},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4735385775566101},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4637297987937927},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1815795600414276},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06911405920982361},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1017/s0963548306008376","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548306008376","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A56C6BF6303133B88B09EB0630A9E845/S0963548306008376a.pdf/div-class-title-first-order-definability-of-trees-and-sparse-random-graphs-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:edoc.hu-berlin.de:18452/28384","is_oa":false,"landing_page_url":"http://edoc.hu-berlin.de/18452/28384","pdf_url":null,"source":{"id":"https://openalex.org/S4306400923","display_name":"edoc Publication server (Humboldt University of Berlin)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I39343248","host_organization_name":"Humboldt-Universit\u00e4t zu Berlin","host_organization_lineage":["https://openalex.org/I39343248"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"},{"id":"doi:10.18452/27701","is_oa":true,"landing_page_url":"https://doi.org/10.18452/27701","pdf_url":null,"source":{"id":"https://openalex.org/S4306400923","display_name":"edoc Publication server (Humboldt University of Berlin)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I39343248","host_organization_name":"Humboldt-Universit\u00e4t zu Berlin","host_organization_lineage":["https://openalex.org/I39343248"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.1017/s0963548306008376","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548306008376","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A56C6BF6303133B88B09EB0630A9E845/S0963548306008376a.pdf/div-class-title-first-order-definability-of-trees-and-sparse-random-graphs-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2238408156","display_name":"Probabilistic Considerations in the Analysis of Algorithms","funder_award_id":"0200945","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320273","display_name":"University of Cambridge","ror":"https://ror.org/013meh722"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2098725746.pdf","grobid_xml":"https://content.openalex.org/works/W2098725746.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W390146837","https://openalex.org/W1564608677","https://openalex.org/W1566493102","https://openalex.org/W1966115134","https://openalex.org/W1986313631","https://openalex.org/W2000115653","https://openalex.org/W2025099804","https://openalex.org/W2032049676","https://openalex.org/W2083365919","https://openalex.org/W2092294035","https://openalex.org/W2100038936","https://openalex.org/W2119414881","https://openalex.org/W2149745907","https://openalex.org/W2179338236","https://openalex.org/W2478722229","https://openalex.org/W2905110430","https://openalex.org/W2914659449","https://openalex.org/W2950522257","https://openalex.org/W2952496718","https://openalex.org/W3144881883","https://openalex.org/W4247549312","https://openalex.org/W4384817650","https://openalex.org/W4394668251"],"related_works":["https://openalex.org/W2759379989","https://openalex.org/W2359046535","https://openalex.org/W2911665669","https://openalex.org/W2032208719","https://openalex.org/W3161230432","https://openalex.org/W1578347593","https://openalex.org/W3027422232","https://openalex.org/W1487201914","https://openalex.org/W2168900540","https://openalex.org/W4367190712"],"abstract_inverted_index":{"Let":[0],"D(G)":[1],"be":[2,27],"the":[3,33,60,79],"smallest":[4],"quantifier":[5],"depth":[6],"of":[7,36,56,63,81,87],"a":[8,15,30,53,64,84],"first-order":[9,39],"formula":[10],"which":[11],"is":[12,52],"true":[13],"for":[14,20,32,83],"graph":[16,66],"G":[17,37,51],"but":[18],"false":[19],"any":[21],"other":[22],"non-isomorphic":[23],"graph.":[24],"This":[25],"can":[26],"viewed":[28],"as":[29,100],"measure":[31],"descriptive":[34],"complexity":[35],"in":[38],"logic.":[40],"We":[41],"show":[42],"that":[43],"almost":[44],"surely":[45],"$D(G)=\\Theta(\\frac{\\ln":[46],"n}{\\ln\\ln":[47],"n})$":[48],",":[49,94],"where":[50],"random":[54,65],"tree":[55,85],"order":[57,88],"n":[58,89],"or":[59],"giant":[61],"component":[62],"$\\C":[67],"G(n,\\frac":[68],"cn)$":[69],"with":[70],"constant":[71],"c":[72],"<1.":[73],"These":[74],"results":[75],"rely":[76],"on":[77],"computing":[78],"maximum":[80,91],"D(T)":[82],"T":[86],"and":[90],"degree":[92],"l":[93],"so":[95],"we":[96],"study":[97],"this":[98],"problem":[99],"well.":[101]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}